LMFDB - Embedded newform 475.3.c.b.151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(151,475)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 13 $ x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			151.1
Root			$-3.60555i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.b.151.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60555i q^{2} +3.60555i q^{3} -9.00000 q^{4} +13.0000 q^{6} +5.00000 q^{7} +18.0278i q^{8} -4.00000 q^{9} +O(q^{10})$	\(q-3.60555i q^{2} +3.60555i q^{3} -9.00000 q^{4} +13.0000 q^{6} +5.00000 q^{7} +18.0278i q^{8} -4.00000 q^{9} -10.0000 q^{11} -32.4500i q^{12} -3.60555i q^{13} -18.0278i q^{14} +29.0000 q^{16} -15.0000 q^{17} +14.4222i q^{18} +(-6.00000 + 18.0278i) q^{19} +18.0278i q^{21} +36.0555i q^{22} -35.0000 q^{23} -65.0000 q^{24} -13.0000 q^{26} +18.0278i q^{27} -45.0000 q^{28} +18.0278i q^{29} -36.0555i q^{31} -32.4500i q^{32} -36.0555i q^{33} +54.0833i q^{34} +36.0000 q^{36} +21.6333i q^{37} +(65.0000 + 21.6333i) q^{38} +13.0000 q^{39} +36.0555i q^{41} +65.0000 q^{42} +20.0000 q^{43} +90.0000 q^{44} +126.194i q^{46} -10.0000 q^{47} +104.561i q^{48} -24.0000 q^{49} -54.0833i q^{51} +32.4500i q^{52} +75.7166i q^{53} +65.0000 q^{54} +90.1388i q^{56} +(-65.0000 - 21.6333i) q^{57} +65.0000 q^{58} +18.0278i q^{59} -40.0000 q^{61} -130.000 q^{62} -20.0000 q^{63} -1.00000 q^{64} -130.000 q^{66} -39.6611i q^{67} +135.000 q^{68} -126.194i q^{69} +108.167i q^{71} -72.1110i q^{72} -105.000 q^{73} +78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000 q^{77} -46.8722i q^{78} -36.0555i q^{79} -101.000 q^{81} +130.000 q^{82} +40.0000 q^{83} -162.250i q^{84} -72.1110i q^{86} -65.0000 q^{87} -180.278i q^{88} -18.0278i q^{91} +315.000 q^{92} +130.000 q^{93} +36.0555i q^{94} +117.000 q^{96} -122.589i q^{97} +86.5332i q^{98} +40.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{4} + 26 q^{6} + 10 q^{7} - 8 q^{9}+O(q^{10}) $ 2 * q - 18 * q^4 + 26 * q^6 + 10 * q^7 - 8 * q^9	$ 2 q - 18 q^{4} + 26 q^{6} + 10 q^{7} - 8 q^{9} - 20 q^{11} + 58 q^{16} - 30 q^{17} - 12 q^{19} - 70 q^{23} - 130 q^{24} - 26 q^{26} - 90 q^{28} + 72 q^{36} + 130 q^{38} + 26 q^{39} + 130 q^{42} + 40 q^{43} + 180 q^{44} - 20 q^{47} - 48 q^{49} + 130 q^{54} - 130 q^{57} + 130 q^{58} - 80 q^{61} - 260 q^{62} - 40 q^{63} - 2 q^{64} - 260 q^{66} + 270 q^{68} - 210 q^{73} + 156 q^{74} + 108 q^{76} - 100 q^{77} - 202 q^{81} + 260 q^{82} + 80 q^{83} - 130 q^{87} + 630 q^{92} + 260 q^{93} + 234 q^{96} + 80 q^{99}+O(q^{100}) $ 2 * q - 18 * q^4 + 26 * q^6 + 10 * q^7 - 8 * q^9 - 20 * q^11 + 58 * q^16 - 30 * q^17 - 12 * q^19 - 70 * q^23 - 130 * q^24 - 26 * q^26 - 90 * q^28 + 72 * q^36 + 130 * q^38 + 26 * q^39 + 130 * q^42 + 40 * q^43 + 180 * q^44 - 20 * q^47 - 48 * q^49 + 130 * q^54 - 130 * q^57 + 130 * q^58 - 80 * q^61 - 260 * q^62 - 40 * q^63 - 2 * q^64 - 260 * q^66 + 270 * q^68 - 210 * q^73 + 156 * q^74 + 108 * q^76 - 100 * q^77 - 202 * q^81 + 260 * q^82 + 80 * q^83 - 130 * q^87 + 630 * q^92 + 260 * q^93 + 234 * q^96 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	3.60555i		−	1.80278i	−0.433013	−	0.901388i	$-0.642549\pi$
$2$		−	3.60555i		−	1.80278i	0.433013	−	0.901388i	$-0.357451\pi$
$3$			3.60555i			1.20185i	0.799305	+	0.600925i	$0.205201\pi$
$3$			3.60555i			1.20185i	−0.799305	+	0.600925i	$0.794799\pi$
$4$	−9.00000			−2.25000
$5$		0			0
$5$		0			0
$6$	13.0000			2.16667
$7$	5.00000			0.714286			0.357143	−	0.934050i	$-0.383751\pi$
$7$	5.00000			0.714286			0.357143	+	0.934050i	$0.383751\pi$
$8$			18.0278i			2.25347i
$9$	−4.00000			−0.444444
$10$		0			0
$11$	−10.0000			−0.909091			−0.454545	−	0.890724i	$-0.650198\pi$
$11$	−10.0000			−0.909091			−0.454545	+	0.890724i	$0.650198\pi$
$12$		−	32.4500i		−	2.70416i
$13$		−	3.60555i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	3.60555i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		−	18.0278i		−	1.28770i
$15$		0			0
$16$	29.0000			1.81250
$17$	−15.0000			−0.882353			−0.441176	−	0.897420i	$-0.645439\pi$
$17$	−15.0000			−0.882353			−0.441176	+	0.897420i	$0.645439\pi$
$18$			14.4222i			0.801234i
$19$	−6.00000	+	18.0278i	−0.315789	+	0.948829i
$19$	−6.00000	+	18.0278i	−0.315789	+	0.948829i
$20$		0			0
$21$			18.0278i			0.858465i
$22$			36.0555i			1.63889i
$23$	−35.0000			−1.52174			−0.760870	−	0.648905i	$-0.775227\pi$
$23$	−35.0000			−1.52174			−0.760870	+	0.648905i	$0.775227\pi$
$24$	−65.0000			−2.70833
$25$		0			0
$26$	−13.0000			−0.500000
$27$			18.0278i			0.667695i
$28$	−45.0000			−1.60714
$29$			18.0278i			0.621647i	0.950468	+	0.310823i	$0.100605\pi$
$29$			18.0278i			0.621647i	−0.950468	+	0.310823i	$0.899395\pi$
$30$		0			0
$31$		−	36.0555i		−	1.16308i	−0.813517	−	0.581541i	$-0.802450\pi$
$31$		−	36.0555i		−	1.16308i	0.813517	−	0.581541i	$-0.197550\pi$
$32$		−	32.4500i		−	1.01406i
$33$		−	36.0555i		−	1.09259i
$34$			54.0833i			1.59068i
$35$		0			0
$36$	36.0000			1.00000
$37$			21.6333i			0.584684i	0.956314	+	0.292342i	$0.0944346\pi$
$37$			21.6333i			0.584684i	−0.956314	+	0.292342i	$0.905565\pi$
$38$	65.0000	+	21.6333i	1.71053	+	0.569298i
$39$	13.0000			0.333333
$40$		0			0
$41$			36.0555i			0.879403i	0.898144	+	0.439701i	$0.144916\pi$
$41$			36.0555i			0.879403i	−0.898144	+	0.439701i	$0.855084\pi$
$42$	65.0000			1.54762
$43$	20.0000			0.465116			0.232558	−	0.972582i	$-0.425290\pi$
$43$	20.0000			0.465116			0.232558	+	0.972582i	$0.425290\pi$
$44$	90.0000			2.04545
$45$		0			0
$46$			126.194i			2.74335i
$47$	−10.0000			−0.212766			−0.106383	−	0.994325i	$-0.533927\pi$
$47$	−10.0000			−0.212766			−0.106383	+	0.994325i	$0.533927\pi$
$48$			104.561i			2.17835i
$49$	−24.0000			−0.489796
$50$		0			0
$51$		−	54.0833i		−	1.06046i
$52$			32.4500i			0.624038i
$53$			75.7166i			1.42861i	0.699832	+	0.714307i	$0.253258\pi$
$53$			75.7166i			1.42861i	−0.699832	+	0.714307i	$0.746742\pi$
$54$	65.0000			1.20370
$55$		0			0
$56$			90.1388i			1.60962i
$57$	−65.0000	−	21.6333i	−1.14035	−	0.379532i
$58$	65.0000			1.12069
$59$			18.0278i			0.305555i	0.988261	+	0.152778i	$0.0488218\pi$
$59$			18.0278i			0.305555i	−0.988261	+	0.152778i	$0.951178\pi$
$60$		0			0
$61$	−40.0000			−0.655738			−0.327869	−	0.944723i	$-0.606330\pi$
$61$	−40.0000			−0.655738			−0.327869	+	0.944723i	$0.606330\pi$
$62$	−130.000			−2.09677
$63$	−20.0000			−0.317460
$64$	−1.00000			−0.0156250
$65$		0			0
$66$	−130.000			−1.96970
$67$		−	39.6611i		−	0.591956i	−0.955195	−	0.295978i	$-0.904354\pi$
$67$		−	39.6611i		−	0.591956i	0.955195	−	0.295978i	$-0.0956455\pi$
$68$	135.000			1.98529
$69$		−	126.194i		−	1.82890i
$70$		0			0
$71$			108.167i			1.52347i	0.647887	+	0.761736i	$0.275653\pi$
$71$			108.167i			1.52347i	−0.647887	+	0.761736i	$0.724347\pi$
$72$		−	72.1110i		−	1.00154i
$73$	−105.000			−1.43836			−0.719178	−	0.694826i	$-0.755481\pi$
$73$	−105.000			−1.43836			−0.719178	+	0.694826i	$0.755481\pi$
$74$	78.0000			1.05405
$75$		0			0
$76$	54.0000	−	162.250i	0.710526	−	2.13487i
$77$	−50.0000			−0.649351
$78$		−	46.8722i		−	0.600925i
$79$		−	36.0555i		−	0.456399i	−0.973614	−	0.228199i	$-0.926716\pi$
$79$		−	36.0555i		−	0.456399i	0.973614	−	0.228199i	$-0.0732838\pi$
$80$		0			0
$81$	−101.000			−1.24691
$82$	130.000			1.58537
$83$	40.0000			0.481928			0.240964	−	0.970534i	$-0.422536\pi$
$83$	40.0000			0.481928			0.240964	+	0.970534i	$0.422536\pi$
$84$		−	162.250i		−	1.93155i
$85$		0			0
$86$		−	72.1110i		−	0.838500i
$87$	−65.0000			−0.747126
$88$		−	180.278i		−	2.04861i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	18.0278i		−	0.198107i
$92$	315.000			3.42391
$93$	130.000			1.39785
$94$			36.0555i			0.383569i
$95$		0			0
$96$	117.000			1.21875
$97$		−	122.589i		−	1.26380i	−0.775049	−	0.631901i	$-0.782275\pi$
$97$		−	122.589i		−	1.26380i	0.775049	−	0.631901i	$-0.217725\pi$
$98$			86.5332i			0.882992i
$99$	40.0000			0.404040
$100$		0			0
$101$	−50.0000			−0.495050			−0.247525	−	0.968882i	$-0.579617\pi$
$101$	−50.0000			−0.495050			−0.247525	+	0.968882i	$0.579617\pi$
$102$	−195.000			−1.91176
$103$		−	57.6888i		−	0.560086i	−0.959988	−	0.280043i	$-0.909651\pi$
$103$		−	57.6888i		−	0.560086i	0.959988	−	0.280043i	$-0.0903487\pi$
$104$	65.0000			0.625000
$105$		0			0
$106$	273.000			2.57547
$107$		−	75.7166i		−	0.707632i	−0.935315	−	0.353816i	$-0.884884\pi$
$107$		−	75.7166i		−	0.707632i	0.935315	−	0.353816i	$-0.115116\pi$
$108$		−	162.250i		−	1.50231i
$109$		−	198.305i		−	1.81931i	−0.415359	−	0.909657i	$-0.636344\pi$
$109$		−	198.305i		−	1.81931i	0.415359	−	0.909657i	$-0.363656\pi$
$110$		0			0
$111$	−78.0000			−0.702703
$112$	145.000			1.29464
$113$			122.589i			1.08486i	0.840102	+	0.542428i	$0.182495\pi$
$113$			122.589i			1.08486i	−0.840102	+	0.542428i	$0.817505\pi$
$114$	−78.0000	+	234.361i	−0.684211	+	2.05580i
$115$		0			0
$116$		−	162.250i		−	1.39871i
$117$			14.4222i			0.123267i
$118$	65.0000			0.550847
$119$	−75.0000			−0.630252
$120$		0			0
$121$	−21.0000			−0.173554
$122$			144.222i			1.18215i
$123$	−130.000			−1.05691
$124$			324.500i			2.61693i
$125$		0			0
$126$			72.1110i			0.572310i
$127$			129.800i			1.02205i	0.859567	+	0.511023i	$0.170733\pi$
$127$			129.800i			1.02205i	−0.859567	+	0.511023i	$0.829267\pi$
$128$		−	126.194i		−	0.985893i
$129$			72.1110i			0.559000i
$130$		0			0
$131$	112.000			0.854962			0.427481	−	0.904024i	$-0.359401\pi$
$131$	112.000			0.854962			0.427481	+	0.904024i	$0.359401\pi$
$132$			324.500i			2.45833i
$133$	−30.0000	+	90.1388i	−0.225564	+	0.677735i
$134$	−143.000			−1.06716
$135$		0			0
$136$		−	270.416i		−	1.98836i
$137$	−125.000			−0.912409			−0.456204	−	0.889875i	$-0.650791\pi$
$137$	−125.000			−0.912409			−0.456204	+	0.889875i	$0.650791\pi$
$138$	−455.000			−3.29710
$139$	50.0000			0.359712			0.179856	−	0.983693i	$-0.442437\pi$
$139$	50.0000			0.359712			0.179856	+	0.983693i	$0.442437\pi$
$140$		0			0
$141$		−	36.0555i		−	0.255713i
$142$	390.000			2.74648
$143$			36.0555i			0.252136i
$144$	−116.000			−0.805556
$145$		0			0
$146$			378.583i			2.59303i
$147$		−	86.5332i		−	0.588661i
$148$		−	194.700i		−	1.31554i
$149$	70.0000			0.469799			0.234899	−	0.972020i	$-0.424524\pi$
$149$	70.0000			0.469799			0.234899	+	0.972020i	$0.424524\pi$
$150$		0			0
$151$			36.0555i			0.238778i	0.992848	+	0.119389i	$0.0380936\pi$
$151$			36.0555i			0.238778i	−0.992848	+	0.119389i	$0.961906\pi$
$152$	−325.000	−	108.167i	−2.13816	−	0.711622i
$153$	60.0000			0.392157
$154$			180.278i			1.17063i
$155$		0			0
$156$	−117.000			−0.750000
$157$	−10.0000			−0.0636943			−0.0318471	−	0.999493i	$-0.510139\pi$
$157$	−10.0000			−0.0636943			−0.0318471	+	0.999493i	$0.510139\pi$
$158$	−130.000			−0.822785
$159$	−273.000			−1.71698
$160$		0			0
$161$	−175.000			−1.08696
$162$			364.161i			2.24791i
$163$	270.000			1.65644			0.828221	−	0.560402i	$-0.189353\pi$
$163$	270.000			1.65644			0.828221	+	0.560402i	$0.189353\pi$
$164$		−	324.500i		−	1.97866i
$165$		0			0
$166$		−	144.222i		−	0.868808i
$167$		−	122.589i		−	0.734064i	−0.930208	−	0.367032i	$-0.880374\pi$
$167$		−	122.589i		−	0.734064i	0.930208	−	0.367032i	$-0.119626\pi$
$168$	−325.000			−1.93452
$169$	156.000			0.923077
$170$		0			0
$171$	24.0000	−	72.1110i	0.140351	−	0.421702i
$172$	−180.000			−1.04651
$173$			122.589i			0.708605i	0.935131	+	0.354303i	$0.115282\pi$
$173$			122.589i			0.708605i	−0.935131	+	0.354303i	$0.884718\pi$
$174$			234.361i			1.34690i
$175$		0			0
$176$	−290.000			−1.64773
$177$	−65.0000			−0.367232
$178$		0			0
$179$			36.0555i			0.201427i	0.994915	+	0.100714i	$0.0321126\pi$
$179$			36.0555i			0.201427i	−0.994915	+	0.100714i	$0.967887\pi$
$180$		0			0
$181$			108.167i			0.597605i	0.954315	+	0.298803i	$0.0965872\pi$
$181$			108.167i			0.597605i	−0.954315	+	0.298803i	$0.903413\pi$
$182$	−65.0000			−0.357143
$183$		−	144.222i		−	0.788099i
$184$		−	630.971i		−	3.42919i
$185$		0			0
$186$		−	468.722i		−	2.52001i
$187$	150.000			0.802139
$188$	90.0000			0.478723
$189$			90.1388i			0.476925i
$190$		0			0
$191$	193.000			1.01047			0.505236	−	0.862981i	$-0.331406\pi$
$191$	193.000			1.01047			0.505236	+	0.862981i	$0.331406\pi$
$192$		−	3.60555i		−	0.0187789i
$193$		−	266.811i		−	1.38244i	−0.722645	−	0.691220i	$-0.757074\pi$
$193$		−	266.811i		−	1.38244i	0.722645	−	0.691220i	$-0.242926\pi$
$194$	−442.000			−2.27835
$195$		0			0
$196$	216.000			1.10204
$197$	−90.0000			−0.456853			−0.228426	−	0.973561i	$-0.573358\pi$
$197$	−90.0000			−0.456853			−0.228426	+	0.973561i	$0.573358\pi$
$198$		−	144.222i		−	0.728394i
$199$	123.000			0.618090			0.309045	−	0.951047i	$-0.399991\pi$
$199$	123.000			0.618090			0.309045	+	0.951047i	$0.399991\pi$
$200$		0			0
$201$	143.000			0.711443
$202$			180.278i			0.892463i
$203$			90.1388i			0.444033i
$204$			486.749i			2.38603i
$205$		0			0
$206$	−208.000			−1.00971
$207$	140.000			0.676329
$208$		−	104.561i		−	0.502697i
$209$	60.0000	−	180.278i	0.287081	−	0.862572i
$210$		0			0
$211$		−	234.361i		−	1.11071i	−0.831612	−	0.555357i	$-0.812581\pi$
$211$		−	234.361i		−	1.11071i	0.831612	−	0.555357i	$-0.187419\pi$
$212$		−	681.449i		−	3.21438i
$213$	−390.000			−1.83099
$214$	−273.000			−1.27570
$215$		0			0
$216$	−325.000			−1.50463
$217$		−	180.278i		−	0.830772i
$218$	−715.000			−3.27982
$219$		−	378.583i		−	1.72869i
$220$		0			0
$221$			54.0833i			0.244721i
$222$			281.233i			1.26682i
$223$			201.911i			0.905430i	0.891655	+	0.452715i	$0.149544\pi$
$223$			201.911i			0.905430i	−0.891655	+	0.452715i	$0.850456\pi$
$224$		−	162.250i		−	0.724329i
$225$		0			0
$226$	442.000			1.95575
$227$		−	255.994i		−	1.12773i	−0.825868	−	0.563864i	$-0.809314\pi$
$227$		−	255.994i		−	1.12773i	0.825868	−	0.563864i	$-0.190686\pi$
$228$	585.000	+	194.700i	2.56579	+	0.853946i
$229$	−160.000			−0.698690			−0.349345	−	0.936994i	$-0.613596\pi$
$229$	−160.000			−0.698690			−0.349345	+	0.936994i	$0.613596\pi$
$230$		0			0
$231$		−	180.278i		−	0.780422i
$232$	−325.000			−1.40086
$233$	270.000			1.15880			0.579399	−	0.815044i	$-0.303287\pi$
$233$	270.000			1.15880			0.579399	+	0.815044i	$0.303287\pi$
$234$	52.0000			0.222222
$235$		0			0
$236$		−	162.250i		−	0.687499i
$237$	130.000			0.548523
$238$			270.416i			1.13620i
$239$	197.000			0.824268			0.412134	−	0.911123i	$-0.364784\pi$
$239$	197.000			0.824268			0.412134	+	0.911123i	$0.364784\pi$
$240$		0			0
$241$			396.611i			1.64569i	0.568268	+	0.822844i	$0.307614\pi$
$241$			396.611i			1.64569i	−0.568268	+	0.822844i	$0.692386\pi$
$242$			75.7166i			0.312878i
$243$		−	201.911i		−	0.830909i
$244$	360.000			1.47541
$245$		0			0
$246$			468.722i			1.90537i
$247$	65.0000	+	21.6333i	0.263158	+	0.0875842i
$248$	650.000			2.62097
$249$			144.222i			0.579205i
$250$		0			0
$251$	−402.000			−1.60159			−0.800797	−	0.598936i	$-0.795590\pi$
$251$	−402.000			−1.60159			−0.800797	+	0.598936i	$0.795590\pi$
$252$	180.000			0.714286
$253$	350.000			1.38340
$254$	468.000			1.84252
$255$		0			0
$256$	−459.000			−1.79297
$257$			418.244i			1.62741i	0.581279	+	0.813704i	$0.302552\pi$
$257$			418.244i			1.62741i	−0.581279	+	0.813704i	$0.697448\pi$
$258$	260.000			1.00775
$259$			108.167i			0.417631i
$260$		0			0
$261$		−	72.1110i		−	0.276287i
$262$		−	403.822i		−	1.54130i
$263$	−310.000			−1.17871			−0.589354	−	0.807875i	$-0.700618\pi$
$263$	−310.000			−1.17871			−0.589354	+	0.807875i	$0.700618\pi$
$264$	650.000			2.46212
$265$		0			0
$266$	325.000	+	108.167i	1.22180	+	0.406641i
$267$		0			0
$268$			356.950i			1.33190i
$269$			108.167i			0.402106i	0.979580	+	0.201053i	$0.0644364\pi$
$269$			108.167i			0.402106i	−0.979580	+	0.201053i	$0.935564\pi$
$270$		0			0
$271$	105.000			0.387454			0.193727	−	0.981055i	$-0.437942\pi$
$271$	105.000			0.387454			0.193727	+	0.981055i	$0.437942\pi$
$272$	−435.000			−1.59926
$273$	65.0000			0.238095
$274$			450.694i			1.64487i
$275$		0			0
$276$			1135.75i			4.11503i
$277$	50.0000			0.180505			0.0902527	−	0.995919i	$-0.471233\pi$
$277$	50.0000			0.180505			0.0902527	+	0.995919i	$0.471233\pi$
$278$		−	180.278i		−	0.648480i
$279$			144.222i			0.516925i
$280$		0			0
$281$		−	288.444i		−	1.02649i	−0.858242	−	0.513246i	$-0.828443\pi$
$281$		−	288.444i		−	1.02649i	0.858242	−	0.513246i	$-0.171557\pi$
$282$	−130.000			−0.460993
$283$	320.000			1.13074			0.565371	−	0.824837i	$-0.308733\pi$
$283$	320.000			1.13074			0.565371	+	0.824837i	$0.308733\pi$
$284$		−	973.499i		−	3.42781i
$285$		0			0
$286$	130.000			0.454545
$287$			180.278i			0.628145i
$288$			129.800i			0.450694i
$289$	−64.0000			−0.221453
$290$		0			0
$291$	442.000			1.51890
$292$	945.000			3.23630
$293$			219.939i			0.750644i	0.926895	+	0.375322i	$0.122468\pi$
$293$			219.939i			0.750644i	−0.926895	+	0.375322i	$0.877532\pi$
$294$	−312.000			−1.06122
$295$		0			0
$296$	−390.000			−1.31757
$297$		−	180.278i		−	0.606995i
$298$		−	252.389i		−	0.846942i
$299$			126.194i			0.422054i
$300$		0			0
$301$	100.000			0.332226
$302$	130.000			0.430464
$303$		−	180.278i		−	0.594975i
$304$	−174.000	+	522.805i	−0.572368	+	1.71975i
$305$		0			0
$306$		−	216.333i		−	0.706971i
$307$		−	237.966i		−	0.775135i	−0.921841	−	0.387567i	$-0.873315\pi$
$307$		−	237.966i		−	0.775135i	0.921841	−	0.387567i	$-0.126685\pi$
$308$	450.000			1.46104
$309$	208.000			0.673139
$310$		0			0
$311$	395.000			1.27010			0.635048	−	0.772472i	$-0.280980\pi$
$311$	395.000			1.27010			0.635048	+	0.772472i	$0.280980\pi$
$312$			234.361i			0.751157i
$313$	−125.000			−0.399361			−0.199681	−	0.979861i	$-0.563990\pi$
$313$	−125.000			−0.399361			−0.199681	+	0.979861i	$0.563990\pi$
$314$			36.0555i			0.114826i
$315$		0			0
$316$			324.500i			1.02690i
$317$		−	3.60555i		−	0.0113740i	−0.999984	−	0.00568699i	$-0.998190\pi$
$317$		−	3.60555i		−	0.0113740i	0.999984	−	0.00568699i	$-0.00181023\pi$
$318$			984.315i			3.09533i
$319$		−	180.278i		−	0.565133i
$320$		0			0
$321$	273.000			0.850467
$322$			630.971i			1.95954i
$323$	90.0000	−	270.416i	0.278638	−	0.837202i
$324$	909.000			2.80556
$325$		0			0
$326$		−	973.499i		−	2.98619i
$327$	715.000			2.18654
$328$	−650.000			−1.98171
$329$	−50.0000			−0.151976
$330$		0			0
$331$		−	198.305i		−	0.599110i	−0.954079	−	0.299555i	$-0.903162\pi$
$331$		−	198.305i		−	0.599110i	0.954079	−	0.299555i	$-0.0968382\pi$
$332$	−360.000			−1.08434
$333$		−	86.5332i		−	0.259860i
$334$	−442.000			−1.32335
$335$		0			0
$336$			522.805i			1.55597i
$337$		−	57.6888i		−	0.171183i	−0.996330	−	0.0855917i	$-0.972722\pi$
$337$		−	57.6888i		−	0.171183i	0.996330	−	0.0855917i	$-0.0272781\pi$
$338$		−	562.466i		−	1.66410i
$339$	−442.000			−1.30383
$340$		0			0
$341$			360.555i			1.05735i
$342$	−260.000	−	86.5332i	−0.760234	−	0.253021i
$343$	−365.000			−1.06414
$344$			360.555i			1.04813i
$345$		0			0
$346$	442.000			1.27746
$347$	40.0000			0.115274			0.0576369	−	0.998338i	$-0.481643\pi$
$347$	40.0000			0.115274			0.0576369	+	0.998338i	$0.481643\pi$
$348$	585.000			1.68103
$349$	98.0000			0.280802			0.140401	−	0.990095i	$-0.455161\pi$
$349$	98.0000			0.280802			0.140401	+	0.990095i	$0.455161\pi$
$350$		0			0
$351$	65.0000			0.185185
$352$			324.500i			0.921874i
$353$	185.000			0.524079			0.262040	−	0.965057i	$-0.415605\pi$
$353$	185.000			0.524079			0.262040	+	0.965057i	$0.415605\pi$
$354$			234.361i			0.662036i
$355$		0			0
$356$		0			0
$357$		−	270.416i		−	0.757469i
$358$	130.000			0.363128
$359$	−225.000			−0.626741			−0.313370	−	0.949631i	$-0.601458\pi$
$359$	−225.000			−0.626741			−0.313370	+	0.949631i	$0.601458\pi$
$360$		0			0
$361$	−289.000	−	216.333i	−0.800554	−	0.599261i
$362$	390.000			1.07735
$363$		−	75.7166i		−	0.208586i
$364$			162.250i			0.445741i
$365$		0			0
$366$	−520.000			−1.42077
$367$	−50.0000			−0.136240			−0.0681199	−	0.997677i	$-0.521700\pi$
$367$	−50.0000			−0.136240			−0.0681199	+	0.997677i	$0.521700\pi$
$368$	−1015.00			−2.75815
$369$		−	144.222i		−	0.390846i
$370$		0			0
$371$			378.583i			1.02044i
$372$	−1170.00			−3.14516
$373$			436.272i			1.16963i	0.811167	+	0.584815i	$0.198833\pi$
$373$			436.272i			1.16963i	−0.811167	+	0.584815i	$0.801167\pi$
$374$		−	540.833i		−	1.44608i
$375$		0			0
$376$		−	180.278i		−	0.479462i
$377$	65.0000			0.172414
$378$	325.000			0.859788
$379$		−	486.749i		−	1.28430i	−0.766579	−	0.642150i	$-0.778043\pi$
$379$		−	486.749i		−	1.28430i	0.766579	−	0.642150i	$-0.221957\pi$
$380$		0			0
$381$	−468.000			−1.22835
$382$		−	695.871i		−	1.82165i
$383$		−	201.911i		−	0.527182i	−0.964634	−	0.263591i	$-0.915093\pi$
$383$		−	201.911i		−	0.527182i	0.964634	−	0.263591i	$-0.0849070\pi$
$384$	455.000			1.18490
$385$		0			0
$386$	−962.000			−2.49223
$387$	−80.0000			−0.206718
$388$			1103.30i			2.84355i
$389$	−478.000			−1.22879			−0.614396	−	0.788998i	$-0.710600\pi$
$389$	−478.000			−1.22879			−0.614396	+	0.788998i	$0.710600\pi$
$390$		0			0
$391$	525.000			1.34271
$392$		−	432.666i		−	1.10374i
$393$			403.822i			1.02754i
$394$			324.500i			0.823603i
$395$		0			0
$396$	−360.000			−0.909091
$397$	−750.000			−1.88917			−0.944584	−	0.328269i	$-0.893535\pi$
$397$	−750.000			−1.88917			−0.944584	+	0.328269i	$0.893535\pi$
$398$		−	443.483i		−	1.11428i
$399$	−325.000	−	108.167i	−0.814536	−	0.271094i
$400$		0			0
$401$		−	288.444i		−	0.719312i	−0.933085	−	0.359656i	$-0.882894\pi$
$401$		−	288.444i		−	0.719312i	0.933085	−	0.359656i	$-0.117106\pi$
$402$		−	515.594i		−	1.28257i
$403$	−130.000			−0.322581
$404$	450.000			1.11386
$405$		0			0
$406$	325.000			0.800493
$407$		−	216.333i		−	0.531531i
$408$	975.000			2.38971
$409$		−	36.0555i		−	0.0881553i	−0.999028	−	0.0440776i	$-0.985965\pi$
$409$		−	36.0555i		−	0.0881553i	0.999028	−	0.0440776i	$-0.0140349\pi$
$410$		0			0
$411$		−	450.694i		−	1.09658i
$412$			519.199i			1.26019i
$413$			90.1388i			0.218254i
$414$		−	504.777i		−	1.21927i
$415$		0			0
$416$	−117.000			−0.281250
$417$			180.278i			0.432320i
$418$	−650.000	−	216.333i	−1.55502	−	0.517543i
$419$	112.000			0.267303			0.133652	−	0.991028i	$-0.457330\pi$
$419$	112.000			0.267303			0.133652	+	0.991028i	$0.457330\pi$
$420$		0			0
$421$			630.971i			1.49874i	0.662149	+	0.749372i	$0.269645\pi$
$421$			630.971i			1.49874i	−0.662149	+	0.749372i	$0.730355\pi$
$422$	−845.000			−2.00237
$423$	40.0000			0.0945626
$424$	−1365.00			−3.21934
$425$		0			0
$426$			1406.16i			3.30086i
$427$	−200.000			−0.468384
$428$			681.449i			1.59217i
$429$	−130.000			−0.303030
$430$		0			0
$431$		−	432.666i		−	1.00387i	−0.864907	−	0.501933i	$-0.832622\pi$
$431$		−	432.666i		−	1.00387i	0.864907	−	0.501933i	$-0.167378\pi$
$432$			522.805i			1.21020i
$433$			735.532i			1.69869i	0.527839	+	0.849345i	$0.323003\pi$
$433$			735.532i			1.69869i	−0.527839	+	0.849345i	$0.676997\pi$
$434$	−650.000			−1.49770
$435$		0			0
$436$			1784.75i			4.09346i
$437$	210.000	−	630.971i	0.480549	−	1.44387i
$438$	−1365.00			−3.11644
$439$			793.221i			1.80688i	0.428712	+	0.903441i	$0.358967\pi$
$439$			793.221i			1.80688i	−0.428712	+	0.903441i	$0.641033\pi$
$440$		0			0
$441$	96.0000			0.217687
$442$	195.000			0.441176
$443$	−670.000			−1.51242			−0.756208	−	0.654332i	$-0.772950\pi$
$443$	−670.000			−1.51242			−0.756208	+	0.654332i	$0.772950\pi$
$444$	702.000			1.58108
$445$		0			0
$446$	728.000			1.63229
$447$			252.389i			0.564628i
$448$	−5.00000			−0.0111607
$449$		−	36.0555i		−	0.0803018i	−0.999194	−	0.0401509i	$-0.987216\pi$
$449$		−	36.0555i		−	0.0803018i	0.999194	−	0.0401509i	$-0.0127839\pi$
$450$		0			0
$451$		−	360.555i		−	0.799457i
$452$		−	1103.30i		−	2.44093i
$453$	−130.000			−0.286976
$454$	−923.000			−2.03304
$455$		0			0
$456$	390.000	−	1171.80i	0.855263	−	2.56975i
$457$	755.000			1.65208			0.826039	−	0.563612i	$-0.190589\pi$
$457$	755.000			1.65208			0.826039	+	0.563612i	$0.190589\pi$
$458$			576.888i			1.25958i
$459$		−	270.416i		−	0.589142i
$460$		0			0
$461$	772.000			1.67462			0.837310	−	0.546728i	$-0.184127\pi$
$461$	772.000			1.67462			0.837310	+	0.546728i	$0.184127\pi$
$462$	−650.000			−1.40693
$463$	350.000			0.755940			0.377970	−	0.925818i	$-0.376622\pi$
$463$	350.000			0.755940			0.377970	+	0.925818i	$0.376622\pi$
$464$			522.805i			1.12673i
$465$		0			0
$466$		−	973.499i		−	2.08905i
$467$	−70.0000			−0.149893			−0.0749465	−	0.997188i	$-0.523879\pi$
$467$	−70.0000			−0.149893			−0.0749465	+	0.997188i	$0.523879\pi$
$468$		−	129.800i		−	0.277350i
$469$		−	198.305i		−	0.422826i
$470$		0			0
$471$		−	36.0555i		−	0.0765510i
$472$	−325.000			−0.688559
$473$	−200.000			−0.422833
$474$		−	468.722i		−	0.988864i
$475$		0			0
$476$	675.000			1.41807
$477$		−	302.866i		−	0.634940i
$478$		−	710.294i		−	1.48597i
$479$	−370.000			−0.772443			−0.386221	−	0.922406i	$-0.626220\pi$
$479$	−370.000			−0.772443			−0.386221	+	0.922406i	$0.626220\pi$
$480$		0			0
$481$	78.0000			0.162162
$482$	1430.00			2.96680
$483$		−	630.971i		−	1.30636i
$484$	189.000			0.390496
$485$		0			0
$486$	−728.000			−1.49794
$487$			519.199i			1.06612i	0.846078	+	0.533059i	$0.178958\pi$
$487$			519.199i			1.06612i	−0.846078	+	0.533059i	$0.821042\pi$
$488$		−	721.110i		−	1.47768i
$489$			973.499i			1.99080i
$490$		0			0
$491$	−632.000			−1.28717			−0.643585	−	0.765375i	$-0.722554\pi$
$491$	−632.000			−1.28717			−0.643585	+	0.765375i	$0.722554\pi$
$492$	1170.00			2.37805
$493$		−	270.416i		−	0.548512i
$494$	78.0000	−	234.361i	0.157895	−	0.474415i
$495$		0			0
$496$		−	1045.61i		−	2.10808i
$497$			540.833i			1.08819i
$498$	520.000			1.04418
$499$	380.000			0.761523			0.380762	−	0.924673i	$-0.375662\pi$
$499$	380.000			0.761523			0.380762	+	0.924673i	$0.375662\pi$
$500$		0			0
$501$	442.000			0.882236
$502$			1449.43i			2.88731i
$503$	45.0000			0.0894632			0.0447316	−	0.998999i	$-0.485757\pi$
$503$	45.0000			0.0894632			0.0447316	+	0.998999i	$0.485757\pi$
$504$		−	360.555i		−	0.715387i
$505$		0			0
$506$		−	1261.94i		−	2.49396i
$507$			562.466i			1.10940i
$508$		−	1168.20i		−	2.29960i
$509$			829.277i			1.62923i	0.580004	+	0.814614i	$0.303051\pi$
$509$			829.277i			1.62923i	−0.580004	+	0.814614i	$0.696949\pi$
$510$		0			0
$511$	−525.000			−1.02740
$512$			1150.17i			2.24643i
$513$	−325.000	−	108.167i	−0.633528	−	0.210851i
$514$	1508.00			2.93385
$515$		0			0
$516$		−	648.999i		−	1.25775i
$517$	100.000			0.193424
$518$	390.000			0.752896
$519$	−442.000			−0.851638
$520$		0			0
$521$			612.944i			1.17648i	0.808688	+	0.588238i	$0.200178\pi$
$521$			612.944i			1.17648i	−0.808688	+	0.588238i	$0.799822\pi$
$522$	−260.000			−0.498084
$523$		−	465.116i		−	0.889323i	−0.895699	−	0.444662i	$-0.853324\pi$
$523$		−	465.116i		−	0.889323i	0.895699	−	0.444662i	$-0.146676\pi$
$524$	−1008.00			−1.92366
$525$		0			0
$526$			1117.72i			2.12494i
$527$			540.833i			1.02625i
$528$		−	1045.61i		−	1.98032i
$529$	696.000			1.31569
$530$		0			0
$531$		−	72.1110i		−	0.135802i
$532$	270.000	−	811.249i	0.507519	−	1.52490i
$533$	130.000			0.243902
$534$		0			0
$535$		0			0
$536$	715.000			1.33396
$537$	−130.000			−0.242086
$538$	390.000			0.724907
$539$	240.000			0.445269
$540$		0			0
$541$	−600.000			−1.10906			−0.554529	−	0.832165i	$-0.687101\pi$
$541$	−600.000			−1.10906			−0.554529	+	0.832165i	$0.687101\pi$
$542$		−	378.583i		−	0.698492i
$543$	−390.000			−0.718232
$544$			486.749i			0.894760i
$545$		0			0
$546$		−	234.361i		−	0.429232i
$547$			598.522i			1.09419i	0.837071	+	0.547095i	$0.184266\pi$
$547$			598.522i			1.09419i	−0.837071	+	0.547095i	$0.815734\pi$
$548$	1125.00			2.05292
$549$	160.000			0.291439
$550$		0			0
$551$	−325.000	−	108.167i	−0.589837	−	0.196310i
$552$	2275.00			4.12138
$553$		−	180.278i		−	0.325999i
$554$		−	180.278i		−	0.325411i
$555$		0			0
$556$	−450.000			−0.809353
$557$	−380.000			−0.682226			−0.341113	−	0.940022i	$-0.610804\pi$
$557$	−380.000			−0.682226			−0.341113	+	0.940022i	$0.610804\pi$
$558$	520.000			0.931900
$559$		−	72.1110i		−	0.129000i
$560$		0			0
$561$			540.833i			0.964051i
$562$	−1040.00			−1.85053
$563$		−	122.589i		−	0.217742i	−0.994056	−	0.108871i	$-0.965276\pi$
$563$		−	122.589i		−	0.217742i	0.994056	−	0.108871i	$-0.0347235\pi$
$564$			324.500i			0.575354i
$565$		0			0
$566$		−	1153.78i		−	2.03847i
$567$	−505.000			−0.890653
$568$	−1950.00			−3.43310
$569$			36.0555i			0.0633665i	0.999498	+	0.0316832i	$0.0100868\pi$
$569$			36.0555i			0.0633665i	−0.999498	+	0.0316832i	$0.989913\pi$
$570$		0			0
$571$	−790.000			−1.38354			−0.691769	−	0.722119i	$-0.743168\pi$
$571$	−790.000			−1.38354			−0.691769	+	0.722119i	$0.743168\pi$
$572$		−	324.500i		−	0.567307i
$573$			695.871i			1.21444i
$574$	650.000			1.13240
$575$		0			0
$576$	4.00000			0.00694444
$577$	−675.000			−1.16984			−0.584922	−	0.811090i	$-0.698875\pi$
$577$	−675.000			−1.16984			−0.584922	+	0.811090i	$0.698875\pi$
$578$			230.755i			0.399231i
$579$	962.000			1.66149
$580$		0			0
$581$	200.000			0.344234
$582$		−	1593.65i		−	2.73824i
$583$		−	757.166i		−	1.29874i
$584$		−	1892.91i		−	3.24129i
$585$		0			0
$586$	793.000			1.35324
$587$	280.000			0.477002			0.238501	−	0.971142i	$-0.423344\pi$
$587$	280.000			0.477002			0.238501	+	0.971142i	$0.423344\pi$
$588$			778.799i			1.32449i
$589$	650.000	+	216.333i	1.10357	+	0.367289i
$590$		0			0
$591$		−	324.500i		−	0.549069i
$592$			627.366i			1.05974i
$593$	−750.000			−1.26476			−0.632378	−	0.774660i	$-0.717921\pi$
$593$	−750.000			−1.26476			−0.632378	+	0.774660i	$0.717921\pi$
$594$	−650.000			−1.09428
$595$		0			0
$596$	−630.000			−1.05705
$597$			443.483i			0.742852i
$598$	455.000			0.760870
$599$			504.777i			0.842700i	0.906898	+	0.421350i	$0.138444\pi$
$599$			504.777i			0.842700i	−0.906898	+	0.421350i	$0.861556\pi$
$600$		0			0
$601$			612.944i			1.01987i	0.860212	+	0.509937i	$0.170331\pi$
$601$			612.944i			1.01987i	−0.860212	+	0.509937i	$0.829669\pi$
$602$		−	360.555i		−	0.598929i
$603$			158.644i			0.263092i
$604$		−	324.500i		−	0.537251i
$605$		0			0
$606$	−650.000			−1.07261
$607$			987.921i			1.62755i	0.581182	+	0.813774i	$0.302590\pi$
$607$			987.921i			1.62755i	−0.581182	+	0.813774i	$0.697410\pi$
$608$	585.000	+	194.700i	0.962171	+	0.320230i
$609$	−325.000			−0.533662
$610$		0			0
$611$			36.0555i			0.0590107i
$612$	−540.000			−0.882353
$613$	1200.00			1.95759			0.978793	−	0.204853i	$-0.0656715\pi$
$613$	1200.00			1.95759			0.978793	+	0.204853i	$0.0656715\pi$
$614$	−858.000			−1.39739
$615$		0			0
$616$		−	901.388i		−	1.46329i
$617$	350.000			0.567261			0.283630	−	0.958934i	$-0.408461\pi$
$617$	350.000			0.567261			0.283630	+	0.958934i	$0.408461\pi$
$618$		−	749.955i		−	1.21352i
$619$	560.000			0.904685			0.452342	−	0.891844i	$-0.350588\pi$
$619$	560.000			0.904685			0.452342	+	0.891844i	$0.350588\pi$
$620$		0			0
$621$		−	630.971i		−	1.01606i
$622$		−	1424.19i		−	2.28970i
$623$		0			0
$624$	377.000			0.604167
$625$		0			0
$626$			450.694i			0.719958i
$627$	650.000	+	216.333i	1.03668	+	0.345029i
$628$	90.0000			0.143312
$629$		−	324.500i		−	0.515898i
$630$		0			0
$631$	−1050.00			−1.66403			−0.832013	−	0.554757i	$-0.812811\pi$
$631$	−1050.00			−1.66403			−0.832013	+	0.554757i	$0.812811\pi$
$632$	650.000			1.02848
$633$	845.000			1.33491
$634$	−13.0000			−0.0205047
$635$		0			0
$636$	2457.00			3.86321
$637$			86.5332i			0.135845i
$638$	−650.000			−1.01881
$639$		−	432.666i		−	0.677099i
$640$		0			0
$641$			1225.89i			1.91246i	0.292615	+	0.956230i	$0.405475\pi$
$641$			1225.89i			1.91246i	−0.292615	+	0.956230i	$0.594525\pi$
$642$		−	984.315i		−	1.53320i
$643$	1030.00			1.60187			0.800933	−	0.598754i	$-0.204337\pi$
$643$	1030.00			1.60187			0.800933	+	0.598754i	$0.204337\pi$
$644$	1575.00			2.44565
$645$		0			0
$646$	−975.000	−	324.500i	−1.50929	−	0.502321i
$647$	−555.000			−0.857805			−0.428903	−	0.903351i	$-0.641100\pi$
$647$	−555.000			−0.857805			−0.428903	+	0.903351i	$0.641100\pi$
$648$		−	1820.80i		−	2.80988i
$649$		−	180.278i		−	0.277777i
$650$		0			0
$651$	650.000			0.998464
$652$	−2430.00			−3.72699
$653$	−50.0000			−0.0765697			−0.0382848	−	0.999267i	$-0.512189\pi$
$653$	−50.0000			−0.0765697			−0.0382848	+	0.999267i	$0.512189\pi$
$654$		−	2577.97i		−	3.94185i
$655$		0			0
$656$			1045.61i			1.59392i
$657$	420.000			0.639269
$658$			180.278i			0.273978i
$659$			198.305i			0.300919i	0.988616	+	0.150459i	$0.0480752\pi$
$659$			198.305i			0.300919i	−0.988616	+	0.150459i	$0.951925\pi$
$660$		0			0
$661$			198.305i			0.300008i	0.988685	+	0.150004i	$0.0479287\pi$
$661$			198.305i			0.300008i	−0.988685	+	0.150004i	$0.952071\pi$
$662$	−715.000			−1.08006
$663$	−195.000			−0.294118
$664$			721.110i			1.08601i
$665$		0			0
$666$	−312.000			−0.468468
$667$		−	630.971i		−	0.945984i
$668$			1103.30i			1.65164i
$669$	−728.000			−1.08819
$670$		0			0
$671$	400.000			0.596125
$672$	585.000			0.870536
$673$		−	598.522i		−	0.889334i	−0.895696	−	0.444667i	$-0.853322\pi$
$673$		−	598.522i		−	0.889334i	0.895696	−	0.444667i	$-0.146678\pi$
$674$	−208.000			−0.308605
$675$		0			0
$676$	−1404.00			−2.07692
$677$		−	68.5055i		−	0.101190i	−0.998719	−	0.0505949i	$-0.983888\pi$
$677$		−	68.5055i		−	0.101190i	0.998719	−	0.0505949i	$-0.0161117\pi$
$678$			1593.65i			2.35052i
$679$		−	612.944i		−	0.902715i
$680$		0			0
$681$	923.000			1.35536
$682$	1300.00			1.90616
$683$			237.966i			0.348413i	0.984709	+	0.174207i	$0.0557361\pi$
$683$			237.966i			0.348413i	−0.984709	+	0.174207i	$0.944264\pi$
$684$	−216.000	+	648.999i	−0.315789	+	0.948829i
$685$		0			0
$686$			1316.03i			1.91841i
$687$		−	576.888i		−	0.839721i
$688$	580.000			0.843023
$689$	273.000			0.396226
$690$		0			0
$691$	−820.000			−1.18669			−0.593343	−	0.804950i	$-0.702192\pi$
$691$	−820.000			−1.18669			−0.593343	+	0.804950i	$0.702192\pi$
$692$		−	1103.30i		−	1.59436i
$693$	200.000			0.288600
$694$		−	144.222i		−	0.207813i
$695$		0			0
$696$		−	1171.80i		−	1.68363i
$697$		−	540.833i		−	0.775944i
$698$		−	353.344i		−	0.506224i
$699$			973.499i			1.39270i
$700$		0			0
$701$	−540.000			−0.770328			−0.385164	−	0.922848i	$-0.625855\pi$
$701$	−540.000			−0.770328			−0.385164	+	0.922848i	$0.625855\pi$
$702$		−	234.361i		−	0.333847i
$703$	−390.000	−	129.800i	−0.554765	−	0.184637i
$704$	10.0000			0.0142045
$705$		0			0
$706$		−	667.027i		−	0.944797i
$707$	−250.000			−0.353607
$708$	585.000			0.826271
$709$	268.000			0.377997			0.188999	−	0.981977i	$-0.439476\pi$
$709$	268.000			0.377997			0.188999	+	0.981977i	$0.439476\pi$
$710$		0			0
$711$			144.222i			0.202844i
$712$		0			0
$713$			1261.94i			1.76991i
$714$	−975.000			−1.36555
$715$		0			0
$716$		−	324.500i		−	0.453212i
$717$			710.294i			0.990647i
$718$			811.249i			1.12987i
$719$	105.000			0.146036			0.0730181	−	0.997331i	$-0.476737\pi$
$719$	105.000			0.146036			0.0730181	+	0.997331i	$0.476737\pi$
$720$		0			0
$721$		−	288.444i		−	0.400061i
$722$	−780.000	+	1042.00i	−1.08033	+	1.44322i
$723$	−1430.00			−1.97787
$724$		−	973.499i		−	1.34461i
$725$		0			0
$726$	−273.000			−0.376033
$727$	−695.000			−0.955983			−0.477992	−	0.878364i	$-0.658635\pi$
$727$	−695.000			−0.955983			−0.477992	+	0.878364i	$0.658635\pi$
$728$	325.000			0.446429
$729$	−181.000			−0.248285
$730$		0			0
$731$	−300.000			−0.410397
$732$			1298.00i			1.77322i
$733$	−160.000			−0.218281			−0.109141	−	0.994026i	$-0.534810\pi$
$733$	−160.000			−0.218281			−0.109141	+	0.994026i	$0.534810\pi$
$734$			180.278i			0.245610i
$735$		0			0
$736$			1135.75i			1.54314i
$737$			396.611i			0.538142i
$738$	−520.000			−0.704607
$739$	1028.00			1.39107			0.695535	−	0.718493i	$-0.255168\pi$
$739$	1028.00			1.39107			0.695535	+	0.718493i	$0.255168\pi$
$740$		0			0
$741$	−78.0000	+	234.361i	−0.105263	+	0.316276i
$742$	1365.00			1.83962
$743$			526.410i			0.708493i	0.935152	+	0.354247i	$0.115263\pi$
$743$			526.410i			0.708493i	−0.935152	+	0.354247i	$0.884737\pi$
$744$			2343.61i			3.15001i
$745$		0			0
$746$	1573.00			2.10858
$747$	−160.000			−0.214190
$748$	−1350.00			−1.80481
$749$		−	378.583i		−	0.505451i
$750$		0			0
$751$			36.0555i			0.0480100i	0.999712	+	0.0240050i	$0.00764176\pi$
$751$			36.0555i			0.0480100i	−0.999712	+	0.0240050i	$0.992358\pi$
$752$	−290.000			−0.385638
$753$		−	1449.43i		−	1.92488i
$754$		−	234.361i		−	0.310823i
$755$		0			0
$756$		−	811.249i		−	1.07308i
$757$	−60.0000			−0.0792602			−0.0396301	−	0.999214i	$-0.512618\pi$
$757$	−60.0000			−0.0792602			−0.0396301	+	0.999214i	$0.512618\pi$
$758$	−1755.00			−2.31530
$759$			1261.94i			1.66264i
$760$		0			0
$761$	−655.000			−0.860710			−0.430355	−	0.902660i	$-0.641612\pi$
$761$	−655.000			−0.860710			−0.430355	+	0.902660i	$0.641612\pi$
$762$			1687.40i			2.21443i
$763$		−	991.527i		−	1.29951i
$764$	−1737.00			−2.27356
$765$		0			0
$766$	−728.000			−0.950392
$767$	65.0000			0.0847458
$768$		−	1654.95i		−	2.15488i
$769$	−185.000			−0.240572			−0.120286	−	0.992739i	$-0.538381\pi$
$769$	−185.000			−0.240572			−0.120286	+	0.992739i	$0.538381\pi$
$770$		0			0
$771$	−1508.00			−1.95590
$772$			2401.30i			3.11049i
$773$			320.894i			0.415128i	0.978221	+	0.207564i	$0.0665536\pi$
$773$			320.894i			0.415128i	−0.978221	+	0.207564i	$0.933446\pi$
$774$			288.444i			0.372667i
$775$		0			0
$776$	2210.00			2.84794
$777$	−390.000			−0.501931
$778$			1723.45i			2.21524i
$779$	−650.000	−	216.333i	−0.834403	−	0.277706i
$780$		0			0
$781$		−	1081.67i		−	1.38497i
$782$		−	1892.91i		−	2.42061i
$783$	−325.000			−0.415070
$784$	−696.000			−0.887755
$785$		0			0
$786$	1456.00			1.85242
$787$		−	68.5055i		−	0.0870463i	−0.999052	−	0.0435232i	$-0.986142\pi$
$787$		−	68.5055i		−	0.0870463i	0.999052	−	0.0435232i	$-0.0138582\pi$
$788$	810.000			1.02792
$789$		−	1117.72i		−	1.41663i
$790$		0			0
$791$			612.944i			0.774897i
$792$			721.110i			0.910493i
$793$			144.222i			0.181869i
$794$			2704.16i			3.40575i
$795$		0			0
$796$	−1107.00			−1.39070
$797$		−	1258.34i		−	1.57884i	−0.613852	−	0.789421i	$-0.710381\pi$
$797$		−	1258.34i		−	1.57884i	0.613852	−	0.789421i	$-0.289619\pi$
$798$	−390.000	+	1171.80i	−0.488722	+	1.46843i
$799$	150.000			0.187735
$800$		0			0
$801$		0			0
$802$	−1040.00			−1.29676
$803$	1050.00			1.30760
$804$	−1287.00			−1.60075
$805$		0			0
$806$			468.722i			0.581541i
$807$	−390.000			−0.483271
$808$		−	901.388i		−	1.11558i
$809$	−7.00000			−0.00865266			−0.00432633	−	0.999991i	$-0.501377\pi$
$809$	−7.00000			−0.00865266			−0.00432633	+	0.999991i	$0.501377\pi$
$810$		0			0
$811$			1027.58i			1.26706i	0.773720	+	0.633528i	$0.218394\pi$
$811$			1027.58i			1.26706i	−0.773720	+	0.633528i	$0.781606\pi$
$812$		−	811.249i		−	0.999075i
$813$			378.583i			0.465662i
$814$	−780.000			−0.958231
$815$		0			0
$816$		−	1568.41i		−	1.92208i
$817$	−120.000	+	360.555i	−0.146879	+	0.441316i
$818$	−130.000			−0.158924
$819$			72.1110i			0.0880477i
$820$		0			0
$821$	842.000			1.02558			0.512789	−	0.858515i	$-0.328612\pi$
$821$	842.000			1.02558			0.512789	+	0.858515i	$0.328612\pi$
$822$	−1625.00			−1.97689
$823$	−775.000			−0.941677			−0.470838	−	0.882219i	$-0.656049\pi$
$823$	−775.000			−0.941677			−0.470838	+	0.882219i	$0.656049\pi$
$824$	1040.00			1.26214
$825$		0			0
$826$	325.000			0.393462
$827$			1265.55i			1.53029i	0.643859	+	0.765144i	$0.277332\pi$
$827$			1265.55i			1.53029i	−0.643859	+	0.765144i	$0.722668\pi$
$828$	−1260.00			−1.52174
$829$		−	1568.41i		−	1.89194i	−0.324260	−	0.945968i	$-0.605115\pi$
$829$		−	1568.41i		−	1.89194i	0.324260	−	0.945968i	$-0.394885\pi$
$830$		0			0
$831$			180.278i			0.216941i
$832$			3.60555i			0.00433360i
$833$	360.000			0.432173
$834$	650.000			0.779376
$835$		0			0
$836$	−540.000	+	1622.50i	−0.645933	+	1.94079i
$837$	650.000			0.776583
$838$		−	403.822i		−	0.481888i
$839$		−	1153.78i		−	1.37518i	−0.726099	−	0.687590i	$-0.758669\pi$
$839$		−	1153.78i		−	1.37518i	0.726099	−	0.687590i	$-0.241331\pi$
$840$		0			0
$841$	516.000			0.613555
$842$	2275.00			2.70190
$843$	1040.00			1.23369
$844$			2109.25i			2.49911i
$845$		0			0
$846$		−	144.222i		−	0.170475i
$847$	−105.000			−0.123967
$848$			2195.78i			2.58936i
$849$			1153.78i			1.35898i
$850$		0			0
$851$		−	757.166i		−	0.889737i
$852$	3510.00			4.11972
$853$	−400.000			−0.468933			−0.234467	−	0.972124i	$-0.575334\pi$
$853$	−400.000			−0.468933			−0.234467	+	0.972124i	$0.575334\pi$
$854$			721.110i			0.844391i
$855$		0			0
$856$	1365.00			1.59463
$857$		−	158.644i		−	0.185116i	−0.995707	−	0.0925579i	$-0.970496\pi$
$857$		−	158.644i		−	0.185116i	0.995707	−	0.0925579i	$-0.0295043\pi$
$858$			468.722i			0.546296i
$859$	1432.00			1.66705			0.833527	−	0.552478i	$-0.186318\pi$
$859$	1432.00			1.66705			0.833527	+	0.552478i	$0.186318\pi$
$860$		0			0
$861$	−650.000			−0.754936
$862$	−1560.00			−1.80974
$863$			129.800i			0.150405i	0.997168	+	0.0752027i	$0.0239604\pi$
$863$			129.800i			0.150405i	−0.997168	+	0.0752027i	$0.976040\pi$
$864$	585.000			0.677083
$865$		0			0
$866$	2652.00			3.06236
$867$		−	230.755i		−	0.266154i
$868$			1622.50i			1.86924i
$869$			360.555i			0.414908i
$870$		0			0
$871$	−143.000			−0.164179
$872$	3575.00			4.09977
$873$			490.355i			0.561690i
$874$	−2275.00	−	757.166i	−2.60297	−	0.866322i
$875$		0			0
$876$			3407.25i			3.88955i
$877$		−	393.005i		−	0.448124i	−0.974575	−	0.224062i	$-0.928068\pi$
$877$		−	393.005i		−	0.448124i	0.974575	−	0.224062i	$-0.0719319\pi$
$878$	2860.00			3.25740
$879$	−793.000			−0.902162
$880$		0			0
$881$	750.000			0.851305			0.425653	−	0.904887i	$-0.360045\pi$
$881$	750.000			0.851305			0.425653	+	0.904887i	$0.360045\pi$
$882$		−	346.133i		−	0.392441i
$883$	−120.000			−0.135900			−0.0679502	−	0.997689i	$-0.521646\pi$
$883$	−120.000			−0.135900			−0.0679502	+	0.997689i	$0.521646\pi$
$884$		−	486.749i		−	0.550622i
$885$		0			0
$886$			2415.72i			2.72655i
$887$		−	850.910i		−	0.959312i	−0.877457	−	0.479656i	$-0.840761\pi$
$887$		−	850.910i		−	0.959312i	0.877457	−	0.479656i	$-0.159239\pi$
$888$		−	1406.16i		−	1.58352i
$889$			648.999i			0.730033i
$890$		0			0
$891$	1010.00			1.13356
$892$		−	1817.20i		−	2.03722i
$893$	60.0000	−	180.278i	0.0671892	−	0.201879i
$894$	910.000			1.01790
$895$		0			0
$896$		−	630.971i		−	0.704209i
$897$	−455.000			−0.507246
$898$	−130.000			−0.144766
$899$	650.000			0.723026
$900$		0			0
$901$		−	1135.75i		−	1.26054i
$902$	−1300.00			−1.44124
$903$			360.555i			0.399286i
$904$	−2210.00			−2.44469
$905$		0			0
$906$			468.722i			0.517353i
$907$			645.394i			0.711570i	0.934568	+	0.355785i	$0.115786\pi$
$907$			645.394i			0.711570i	−0.934568	+	0.355785i	$0.884214\pi$
$908$			2303.95i			2.53739i
$909$	200.000			0.220022
$910$		0			0
$911$		−	1225.89i		−	1.34565i	−0.739802	−	0.672825i	$-0.765081\pi$
$911$		−	1225.89i		−	1.34565i	0.739802	−	0.672825i	$-0.234919\pi$
$912$	−1885.00	−	627.366i	−2.06689	−	0.687901i
$913$	−400.000			−0.438116
$914$		−	2722.19i		−	2.97833i
$915$		0			0
$916$	1440.00			1.57205
$917$	560.000			0.610687
$918$	−975.000			−1.06209
$919$	−513.000			−0.558215			−0.279108	−	0.960260i	$-0.590039\pi$
$919$	−513.000			−0.558215			−0.279108	+	0.960260i	$0.590039\pi$
$920$		0			0
$921$	858.000			0.931596
$922$		−	2783.49i		−	3.01896i
$923$	390.000			0.422535
$924$			1622.50i			1.75595i
$925$		0			0
$926$		−	1261.94i		−	1.36279i
$927$			230.755i			0.248927i
$928$	585.000			0.630388
$929$	−1217.00			−1.31001			−0.655005	−	0.755624i	$-0.727334\pi$
$929$	−1217.00			−1.31001			−0.655005	+	0.755624i	$0.727334\pi$
$930$		0			0
$931$	144.000	−	432.666i	0.154672	−	0.464733i
$932$	−2430.00			−2.60730
$933$			1424.19i			1.52647i
$934$			252.389i			0.270223i
$935$		0			0
$936$	−260.000			−0.277778
$937$	1095.00			1.16862			0.584312	−	0.811529i	$-0.301365\pi$
$937$	1095.00			1.16862			0.584312	+	0.811529i	$0.301365\pi$
$938$	−715.000			−0.762260
$939$		−	450.694i		−	0.479972i
$940$		0			0
$941$			1099.69i			1.16864i	0.811522	+	0.584322i	$0.198639\pi$
$941$			1099.69i			1.16864i	−0.811522	+	0.584322i	$0.801361\pi$
$942$	−130.000			−0.138004
$943$		−	1261.94i		−	1.33822i
$944$			522.805i			0.553819i
$945$		0			0
$946$			721.110i			0.762273i
$947$	−640.000			−0.675818			−0.337909	−	0.941179i	$-0.609720\pi$
$947$	−640.000			−0.675818			−0.337909	+	0.941179i	$0.609720\pi$
$948$	−1170.00			−1.23418
$949$			378.583i			0.398928i
$950$		0			0
$951$	13.0000			0.0136698
$952$		−	1352.08i		−	1.42025i
$953$		−	1168.20i		−	1.22581i	−0.790156	−	0.612906i	$-0.790000\pi$
$953$		−	1168.20i		−	1.22581i	0.790156	−	0.612906i	$-0.210000\pi$
$954$	−1092.00			−1.14465
$955$		0			0
$956$	−1773.00			−1.85460
$957$	650.000			0.679206
$958$			1334.05i			1.39254i
$959$	−625.000			−0.651721
$960$		0			0
$961$	−339.000			−0.352758
$962$		−	281.233i		−	0.292342i
$963$			302.866i			0.314503i
$964$		−	3569.50i		−	3.70280i
$965$		0			0
$966$	−2275.00			−2.35507
$967$	1010.00			1.04447			0.522234	−	0.852802i	$-0.325099\pi$
$967$	1010.00			1.04447			0.522234	+	0.852802i	$0.325099\pi$
$968$		−	378.583i		−	0.391098i
$969$	975.000	+	324.500i	1.00619	+	0.334881i
$970$		0			0
$971$		−	757.166i		−	0.779779i	−0.920862	−	0.389890i	$-0.872513\pi$
$971$		−	757.166i		−	0.779779i	0.920862	−	0.389890i	$-0.127487\pi$
$972$			1817.20i			1.86955i
$973$	250.000			0.256937
$974$	1872.00			1.92197
$975$		0			0
$976$	−1160.00			−1.18852
$977$		−	598.522i		−	0.612612i	−0.951933	−	0.306306i	$-0.900907\pi$
$977$		−	598.522i		−	0.612612i	0.951933	−	0.306306i	$-0.0990930\pi$
$978$	3510.00			3.58896
$979$		0			0
$980$		0			0
$981$			793.221i			0.808584i
$982$			2278.71i			2.32048i
$983$		−	57.6888i		−	0.0586865i	−0.999569	−	0.0293432i	$-0.990658\pi$
$983$		−	57.6888i		−	0.0586865i	0.999569	−	0.0293432i	$-0.00934159\pi$
$984$		−	2343.61i		−	2.38172i
$985$		0			0
$986$	−975.000			−0.988844
$987$		−	180.278i		−	0.182652i
$988$	−585.000	−	194.700i	−0.592105	−	0.197065i
$989$	−700.000			−0.707786
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−1170.00			−1.17944
$993$	715.000			0.720040
$994$	1950.00			1.96177
$995$		0			0
$996$		−	1298.00i		−	1.30321i
$997$	170.000			0.170512			0.0852558	−	0.996359i	$-0.472829\pi$
$997$	170.000			0.170512			0.0852558	+	0.996359i	$0.472829\pi$
$998$		−	1370.11i		−	1.37286i
$999$	−390.000			−0.390390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.c.b.151.1		2
5.2	odd	4		475.3.d.b.474.4		4
5.3	odd	4		475.3.d.b.474.1		4
5.4	even	2		19.3.b.b.18.2	yes	2
15.14	odd	2		171.3.c.b.37.1		2
19.18	odd	2	inner	475.3.c.b.151.2		2
20.19	odd	2		304.3.e.d.113.2		2
40.19	odd	2		1216.3.e.h.1025.1		2
40.29	even	2		1216.3.e.g.1025.2		2
60.59	even	2		2736.3.o.d.721.2		2
95.4	even	18		361.3.f.d.307.1		12
95.9	even	18		361.3.f.d.299.1		12
95.14	odd	18		361.3.f.d.127.1		12
95.18	even	4		475.3.d.b.474.3		4
95.24	even	18		361.3.f.d.127.2		12
95.29	odd	18		361.3.f.d.299.2		12
95.34	odd	18		361.3.f.d.307.2		12
95.37	even	4		475.3.d.b.474.2		4
95.44	even	18		361.3.f.d.116.2		12
95.49	even	6		361.3.d.b.69.2		4
95.54	even	18		361.3.f.d.333.1		12
95.59	odd	18		361.3.f.d.262.1		12
95.64	even	6		361.3.d.b.293.1		4
95.69	odd	6		361.3.d.b.293.2		4
95.74	even	18		361.3.f.d.262.2		12
95.79	odd	18		361.3.f.d.333.2		12
95.84	odd	6		361.3.d.b.69.1		4
95.89	odd	18		361.3.f.d.116.1		12
95.94	odd	2		19.3.b.b.18.1	✓	2
285.284	even	2		171.3.c.b.37.2		2
380.379	even	2		304.3.e.d.113.1		2
760.189	odd	2		1216.3.e.g.1025.1		2
760.379	even	2		1216.3.e.h.1025.2		2
1140.1139	odd	2		2736.3.o.d.721.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.b.18.1	✓	2	95.94	odd	2
19.3.b.b.18.2	yes	2	5.4	even	2
171.3.c.b.37.1		2	15.14	odd	2
171.3.c.b.37.2		2	285.284	even	2
304.3.e.d.113.1		2	380.379	even	2
304.3.e.d.113.2		2	20.19	odd	2
361.3.d.b.69.1		4	95.84	odd	6
361.3.d.b.69.2		4	95.49	even	6
361.3.d.b.293.1		4	95.64	even	6
361.3.d.b.293.2		4	95.69	odd	6
361.3.f.d.116.1		12	95.89	odd	18
361.3.f.d.116.2		12	95.44	even	18
361.3.f.d.127.1		12	95.14	odd	18
361.3.f.d.127.2		12	95.24	even	18
361.3.f.d.262.1		12	95.59	odd	18
361.3.f.d.262.2		12	95.74	even	18
361.3.f.d.299.1		12	95.9	even	18
361.3.f.d.299.2		12	95.29	odd	18
361.3.f.d.307.1		12	95.4	even	18
361.3.f.d.307.2		12	95.34	odd	18
361.3.f.d.333.1		12	95.54	even	18
361.3.f.d.333.2		12	95.79	odd	18
475.3.c.b.151.1		2	1.1	even	1	trivial
475.3.c.b.151.2		2	19.18	odd	2	inner
475.3.d.b.474.1		4	5.3	odd	4
475.3.d.b.474.2		4	95.37	even	4
475.3.d.b.474.3		4	95.18	even	4
475.3.d.b.474.4		4	5.2	odd	4
1216.3.e.g.1025.1		2	760.189	odd	2
1216.3.e.g.1025.2		2	40.29	even	2
1216.3.e.h.1025.1		2	40.19	odd	2
1216.3.e.h.1025.2		2	760.379	even	2
2736.3.o.d.721.1		2	1140.1139	odd	2
2736.3.o.d.721.2		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.60555i q^{2} +3.60555i q^{3} -9.00000 q^{4} +13.0000 q^{6} +5.00000 q^{7} +18.0278i q^{8} -4.00000 q^{9} +O(q^{10})\)	\(q-3.60555i q^{2} +3.60555i q^{3} -9.00000 q^{4} +13.0000 q^{6} +5.00000 q^{7} +18.0278i q^{8} -4.00000 q^{9} -10.0000 q^{11} -32.4500i q^{12} -3.60555i q^{13} -18.0278i q^{14} +29.0000 q^{16} -15.0000 q^{17} +14.4222i q^{18} +(-6.00000 + 18.0278i) q^{19} +18.0278i q^{21} +36.0555i q^{22} -35.0000 q^{23} -65.0000 q^{24} -13.0000 q^{26} +18.0278i q^{27} -45.0000 q^{28} +18.0278i q^{29} -36.0555i q^{31} -32.4500i q^{32} -36.0555i q^{33} +54.0833i q^{34} +36.0000 q^{36} +21.6333i q^{37} +(65.0000 + 21.6333i) q^{38} +13.0000 q^{39} +36.0555i q^{41} +65.0000 q^{42} +20.0000 q^{43} +90.0000 q^{44} +126.194i q^{46} -10.0000 q^{47} +104.561i q^{48} -24.0000 q^{49} -54.0833i q^{51} +32.4500i q^{52} +75.7166i q^{53} +65.0000 q^{54} +90.1388i q^{56} +(-65.0000 - 21.6333i) q^{57} +65.0000 q^{58} +18.0278i q^{59} -40.0000 q^{61} -130.000 q^{62} -20.0000 q^{63} -1.00000 q^{64} -130.000 q^{66} -39.6611i q^{67} +135.000 q^{68} -126.194i q^{69} +108.167i q^{71} -72.1110i q^{72} -105.000 q^{73} +78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000 q^{77} -46.8722i q^{78} -36.0555i q^{79} -101.000 q^{81} +130.000 q^{82} +40.0000 q^{83} -162.250i q^{84} -72.1110i q^{86} -65.0000 q^{87} -180.278i q^{88} -18.0278i q^{91} +315.000 q^{92} +130.000 q^{93} +36.0555i q^{94} +117.000 q^{96} -122.589i q^{97} +86.5332i q^{98} +40.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{4} + 26 q^{6} + 10 q^{7} - 8 q^{9}+O(q^{10}) \) 2 * q - 18 * q^4 + 26 * q^6 + 10 * q^7 - 8 * q^9	\( 2 q - 18 q^{4} + 26 q^{6} + 10 q^{7} - 8 q^{9} - 20 q^{11} + 58 q^{16} - 30 q^{17} - 12 q^{19} - 70 q^{23} - 130 q^{24} - 26 q^{26} - 90 q^{28} + 72 q^{36} + 130 q^{38} + 26 q^{39} + 130 q^{42} + 40 q^{43} + 180 q^{44} - 20 q^{47} - 48 q^{49} + 130 q^{54} - 130 q^{57} + 130 q^{58} - 80 q^{61} - 260 q^{62} - 40 q^{63} - 2 q^{64} - 260 q^{66} + 270 q^{68} - 210 q^{73} + 156 q^{74} + 108 q^{76} - 100 q^{77} - 202 q^{81} + 260 q^{82} + 80 q^{83} - 130 q^{87} + 630 q^{92} + 260 q^{93} + 234 q^{96} + 80 q^{99}+O(q^{100}) \) 2 * q - 18 * q^4 + 26 * q^6 + 10 * q^7 - 8 * q^9 - 20 * q^11 + 58 * q^16 - 30 * q^17 - 12 * q^19 - 70 * q^23 - 130 * q^24 - 26 * q^26 - 90 * q^28 + 72 * q^36 + 130 * q^38 + 26 * q^39 + 130 * q^42 + 40 * q^43 + 180 * q^44 - 20 * q^47 - 48 * q^49 + 130 * q^54 - 130 * q^57 + 130 * q^58 - 80 * q^61 - 260 * q^62 - 40 * q^63 - 2 * q^64 - 260 * q^66 + 270 * q^68 - 210 * q^73 + 156 * q^74 + 108 * q^76 - 100 * q^77 - 202 * q^81 + 260 * q^82 + 80 * q^83 - 130 * q^87 + 630 * q^92 + 260 * q^93 + 234 * q^96 + 80 * q^99

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